Estimators for the difference in means/medians (2 related samples)

Cohen's d is the most popular estimator using the difference between the means divided by the pooled sample standard deviation. Cohen's d is a biased estimator of the population standardized mean difference (although the bias is very small and disappears for moderate to large samples) whereas Hedge's g applies an unbiasing constant to correct for the bias.

A shift in location is equivalent to a difference between means/medians when the distribution of the differences is symmetric. Symmetry is quite often inherently satisfied for paired data.